Classifying bicrossed products of two Sweedler's Hopf algebras

@article{Bontea2014,
abstract = {We continue the study started recently by Agore, Bontea and Militaru in “Classifying bicrossed products of Hopf algebras” (2014), by describing and classifying all Hopf algebras $E$ that factorize through two Sweedler’s Hopf algebras. Equivalently, we classify all bicrossed products $H_4 \bowtie H_4$. There are three steps in our approach. First, we explicitly describe the set of all matched pairs $(H_4, H_4, \triangleright , \triangleleft )$ by proving that, with the exception of the trivial pair, this set is parameterized by the ground field $k$. Then, for any $\lambda \in k$, we describe by generators and relations the associated bicrossed product, $\mathcal \{H\}_\{16, \lambda \}$. This is a $16$-dimensional, pointed, unimodular and non-semisimple Hopf algebra. A Hopf algebra $E$ factorizes through $H_4$ and $H_4$ if and only if $ E \cong H_4 \otimes H_4$ or $E \cong \{\mathcal \{H\}\}_\{16, \lambda \}$. In the last step we classify these quantum groups by proving that there are only three isomorphism classes represented by: $H_4 \otimes H_4$, $\{\mathcal \{H\}\}_\{16, 0\}$ and $\{\mathcal \{H\}\}_\{16, 1\} \cong D(H_4)$, the Drinfel’d double of $H_4$. The automorphism group of these objects is also computed: in particular, we prove that $\{\rm Aut\}_\{\rm Hopf\}( D(H_4))$ is isomorphic to a semidirect product of groups, $k^\{\times \} \rtimes \mathbb \{Z\}_2$.},
author = {Bontea, Costel-Gabriel},
journal = {Czechoslovak Mathematical Journal},
keywords = {bicrossed product of Hopf algebras; Sweedler's Hopf algebra; Drinfel'd double; factorization problem for Hopf algebras; bicrossed products of Hopf algebras; Sweedler Hopf algebras; quantum groups; Drinfel'd doubles; coalgebras; Hopf algebra automorphisms},
language = {eng},
number = {2},
pages = {419-431},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Classifying bicrossed products of two Sweedler's Hopf algebras},
url = {http://eudml.org/doc/262008},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Bontea, Costel-Gabriel
TI - Classifying bicrossed products of two Sweedler's Hopf algebras
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 2
SP - 419
EP - 431
AB - We continue the study started recently by Agore, Bontea and Militaru in “Classifying bicrossed products of Hopf algebras” (2014), by describing and classifying all Hopf algebras $E$ that factorize through two Sweedler’s Hopf algebras. Equivalently, we classify all bicrossed products $H_4 \bowtie H_4$. There are three steps in our approach. First, we explicitly describe the set of all matched pairs $(H_4, H_4, \triangleright , \triangleleft )$ by proving that, with the exception of the trivial pair, this set is parameterized by the ground field $k$. Then, for any $\lambda \in k$, we describe by generators and relations the associated bicrossed product, $\mathcal {H}_{16, \lambda }$. This is a $16$-dimensional, pointed, unimodular and non-semisimple Hopf algebra. A Hopf algebra $E$ factorizes through $H_4$ and $H_4$ if and only if $ E \cong H_4 \otimes H_4$ or $E \cong {\mathcal {H}}_{16, \lambda }$. In the last step we classify these quantum groups by proving that there are only three isomorphism classes represented by: $H_4 \otimes H_4$, ${\mathcal {H}}_{16, 0}$ and ${\mathcal {H}}_{16, 1} \cong D(H_4)$, the Drinfel’d double of $H_4$. The automorphism group of these objects is also computed: in particular, we prove that ${\rm Aut}_{\rm Hopf}( D(H_4))$ is isomorphic to a semidirect product of groups, $k^{\times } \rtimes \mathbb {Z}_2$.
LA - eng
KW - bicrossed product of Hopf algebras; Sweedler's Hopf algebra; Drinfel'd double; factorization problem for Hopf algebras; bicrossed products of Hopf algebras; Sweedler Hopf algebras; quantum groups; Drinfel'd doubles; coalgebras; Hopf algebra automorphisms
UR - http://eudml.org/doc/262008
ER -